I don't understand how the two notations for $SO(n)$ heighest weights in fact 1 and 2 are related; moreover, what are the weigths of all other irreducible spinorial representations in notation 2?
Fact 1. Define coroots $\alpha^\vee=\frac{2\alpha}{(\alpha,\alpha)}$ and fundamental weights $\Lambda_{(i)}(\alpha^{(j)\vee})=\delta^i_j$, and write a weight as integer linear combination $\Lambda=\sum_{i=1}^r \Lambda^i \Lambda_{(i)}$. Then for $n=2r+1$, $\Lambda^r \pmod 2$ and for $n=2r$, $\Lambda^{r-1} -\Lambda^r \pmod 2$ distinguish heighest weights of tensor and spinor representations.
Fact 2. Clifford algebra has representations $S$ with weights $(\pm \frac12,\ldots,\pm \frac12)$ and heighest weghts $(\frac12,\ldots,\frac12,+\frac12)$ and $(\frac12,\ldots,\frac12,-\frac12)$ for even $n$, corresponding to $\Lambda_{n/2-1}$ and $\Lambda_{n/2}$ respectively, while $(\frac12,\ldots,\frac12)$ for odd $n$, corresponding to $\Lambda_{(n-1)/2}$. More in general, this second notation seems to be: pick Cartan sub algebra generators $H_i$ and dual generators $L_i$ such that $<L_i,H_j>=\delta_{ij}$ and dominant weights are of the form $\lambda = \sum_i \lambda_i L_i$ with $\lambda_i \geq \lambda_{i+1} \geq 0$ for odd $n$ and $\lambda_1 \geq \ldots \geq \lambda_{r-1} \geq |\lambda_r|$ for even $n$.